Mean-Variance Optimization

# Wn 2 i j 1 n st w i 1 i i 0 i 1 n w i 1 where w1

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Unformatted text preview: j w1 ,...,wN 2 i , j =1 N s.t. ∑μw i =1 i i = μ0 i =1 N ∑w i =1 where w1,..., wN : portfolio weights μ1,..., μN : expected returns σij : covariance between asset i and asset j μ0 : desired level of portfolio return RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 3 Here is the Situation: How to Think About the Problem Intuitively RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 4 Vector Notation: Markowitz’s Mean-Variance Framework An investor allocates his wealth by solving the portfolio optimization problem Problem: 1 min w ′Σw w2 s.t. w ′μ = μ0 w ′ι = 1 ′ ι = ⎡⎢⎣1,...,1⎤⎥⎦ where w: portfolio weights μ : vector of expected returns Σ : covariance matrix of returns μ0 : desired level of portfolio return RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 5 How Do We Solve This Problem Mathematically? • With no constraints and/or with linear equality constraints (as stated above) (i.e. Aeq w = beq ) → multivariate calculus gives us...
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